#323567

order	512
nilpotency class	3
isoclinism	isoclinic class of #298867
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

608 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,5,8)(3,7,4,6)(9,10,14,22,13,16,12,17)(11,18,21,15,20,24,19,23)(25,26,30,32,29,33,28,34)(27,31,35,40,36,38,37,39);(1,3,5,4)(2,6,8,7)(25,27,28,37,29,36,30,35)(26,31,34,39,33,38,32,40);(9,11,13,20)(10,15,16,23)(12,21,14,19)(17,24,22,18)(26,32)(27,35)(28,30)(31,38)(33,34)(36,37);(1,4,5,3)(2,7,8,6)(10,16)(15,23)(17,22)(18,24)(26,33)(27,36)(32,34)(35,37)

Polycyclic presentation:

2^9|a^2=efh,b^2=eg,c^2=hi,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,d^b=di,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 230763);